Problem: Simplify and expand the following expression: $ \dfrac{p + 1}{p - 1}+\dfrac{p + 6}{2p - 3} $
Solution: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(p - 1)(2p - 3)$ Multiply the first term by $\dfrac{2p - 3}{2p - 3}$ $ \begin{align*} \dfrac{p + 1}{p - 1} \times \dfrac{2p - 3}{2p - 3} & = \dfrac{(p + 1)(2p - 3)}{(p - 1)(2p - 3)} \\ & = \dfrac{2p^2 - p - 3}{(p - 1)(2p - 3)}\end{align*} $ Multiply the second term by $\dfrac{p - 1}{p - 1}$ $ \begin{align*} \dfrac{p + 6}{2p - 3} \times \dfrac{p - 1}{p - 1} & = \dfrac{(p + 6)(p - 1)}{(2p - 3)(p - 1)} \\ & = \dfrac{p^2 + 5p - 6}{(2p - 3)(p - 1)}\end{align*} $ Now we have: $ = \dfrac{2p^2 - p - 3}{(p - 1)(2p - 3)} + \dfrac{p^2 + 5p - 6}{(2p - 3)(p - 1)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{2p^2 - p - 3 + p^2 + 5p - 6}{(p - 1)(2p - 3)} $ $ = \dfrac{3p^2 + 4p - 9}{(p - 1)(2p - 3)}$ Expand the denominator: $ = \dfrac{3p^2 + 4p - 9}{2p^2 - 5p + 3}$